Nonstandard analysis and locally compact Abelian groups

Using nonstandard analysis, a new method is introduced for locally compact Abelian topological groups. Hyperfiniteness plays an important role. All LCA groups can be obtained this way. This yields approximation results for the (generalized) Fourier transform.     

Limit theorems on locally compact Abelian groups

We prove limit theorems for row sums of a rowwise independent infinitesimal array of random variables with values in a locally compact Abelian group. First we give a proof of Gaiser's theorem [4, Satz 1.3.6], since it does not have an easy access and it is not complete. This theorem gives sufficient conditions for convergence of the row sums, but the limit measure cannot have a nondegenerate idempotent factor. Then we prove necessary and sufficient conditions for convergence of the row sums, where the limit measure can be also a nondegenerate Haar measure on a compact subgroup. Finally, we investigate special cases: the torus group, the group of p ‐adic integers and the p ‐adic solenoid. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Heyde theorem for some locally compact Abelian groups

According to the well-known Heyde theorem Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form in n independent random variables given another. For n = 2 we prove analogs of this theorem in the case when random variables take values in a locally compact Abelian group X, and coefficients of the linear forms are topological automorphisms of the group X.     

On spectral analysis in locally compact motion groups

Beurling's theorem in spectral analysis of bounded functions on the real line is generalized to a class of locally compact motion groups and to Heisenberg groups.     

Mixing rank-one actions of locally compact Abelian groups

Using techniques related to the (C,F)-actions we construct explicitly mixing rank-one (by cubes) actions T of G=Rd₁×Zd₂ for any pair of non-negative integers d₁, d₂. It is also shown that h(Tg)=0 for each g∈G.     

On some algebraic properties of locally compact and weakly linearly compact topological Abelian groups

Locally compact and weakly linearly compact topological groups are studied. The notion of a weakly linearly compact topological Abelian group is a generalization of the notion of a weakly separable topological Abelian group, introduced by N. Ya. Vilenkin. Some algebraic properties of these groups are studied. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 41, Topology and Its Applications, 2006.     

The Heyde theorem for locally compact Abelian groups

We prove a group analogue of the well-known Heyde theorem where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to the circle group T, G be the subgroup of X generated by all elements of order 2, and Aut(X) be the set of all topological automorphisms of X. Let αj,βj∈Aut(X), j=1,2,…,n, n?2, such that for all i≠j. Let ξj be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. If the conditional distribution of L₂=β₁ξ₁+?+βnξn given L₁=α₁ξ₁+?+αnξn is symmetric, then each μj=γj∗ρj, where γj are Gaussian measures, and ρj are distributions supported in G.     

Persistently unbounded functions on locally compact Abelian groups

This paper deals, given an arbitrary locally compact Abelian group, with the existence of integrable functions all whose convolution powers are essentially unbounded.     

Mbekhta's subspaces and a spectral theory of compact operators

Let be an operator on an infinite-dimensional complex Banach space. By means of Mbekhta's subspaces and , we give a spectral theory of compact operators. The main results are: Let be compact. . The following assertions are all equivalent: (1) 0 is an isolated point in the spectrum of (2) is closed; (3) is of finite dimension; (4) is closed; (5) is of finite dimension; . sufficient conditions for to be an isolated point in ; . sufficient and necessary conditions for to be a pole of the resolvent of .

     

On locally free Abelian groups

The following result is proved in the paper. An Abelian group A is Lw₁, w-equivalent to the free Abelian group of countable rank if and only if it is a countably free Abelian group.     

On locally free Abelian groups

The following result is proved in the paper. An Abelian group A is Lw₁, w-equivalent to the free Abelian group of countable rank if and only if it is a countably free Abelian group.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 121–126.Original Russian Text Copyright © 2005 by E. G. Sklyarenko.This revised version was published online in April 2005 with a corrected issue number.